Exercise 1.1 answers: Sets

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(i) all the months of a year beginning with the letter J

1. The months of a year starting with J are January, June and July.
2. Anyone checking any month gets the same yes/no answer, so the test is fixed.

AnswerYes, it is a set.

(ii) ten most talented writers of India

1. "Most talented" needs a personal ranking − two people can rank writers differently.
2. Since the test is not the same for everyone, the collection has no fixed test.

AnswerNo, it is not a set.

(iii) eleven best-cricket batsmen of the world

1. "Best" batsmen depends on whose measure is used − by average, by matches won, or something else.
2. Two selectors could honestly name two different elevens, so there is no single fixed test.

AnswerNo, it is not a set.

(iv) all boys in your class

1. Every student is either on your class register or not.
2. Anyone can check the register and get the same answer for any boy.

AnswerYes, it is a set.

(v) all natural numbers less than 100

1. A natural number is either less than 100 or it is not, and this can always be checked exactly.

AnswerYes, it is a set.

(vi) novels written by the writer Munshi Prem Chand

1. Munshi Prem Chand is one fixed author, so a novel either carries his name or it does not.

AnswerYes, it is a set.

(vii) all even integers

1. Any integer is either divisible by 2 or it is not, and this test never changes.

AnswerYes, it is a set.

(viii) questions in this Chapter

1. The questions printed in this chapter are a fixed, countable list.

AnswerYes, it is a set.

(ix) most dangerous animals of the world

1. "Most dangerous" depends on the measure used − speed, number of attacks, venom, and so on.
2. Different people can honestly disagree, so there is no single fixed test.

AnswerNo, it is not a set.

Watch this explained “Size is not the issue”, 9:06 into Why "well-defined" is the whole difference between a set and a heap

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(i) 5 . . . A

1. A = {1,2,3,4,5,6}, and 5 is in this list.

Answer5 ∈ A

(ii) 8 . . . A

1. 8 does not appear in {1,2,3,4,5,6}.

Answer8 ∉ A

(iii) 0 . . . A

1. 0 does not appear in the list.

Answer0 ∉ A

(iv) 4 . . . A

1. 4 is in the list.

Answer4 ∈ A

(v) 2 . . . A

1. 2 is in the list.

Answer2 ∈ A

(vi) 10 . . . A

1. 10 does not appear in the list.

Answer10 ∉ A

Watch this explained “The symbol is a receipt”, 6:31 into Why "well-defined" is the whole difference between a set and a heap

Open NCERT p. 5Matches NCERT’s answer

(i) x is an integer and −3 ≤ x < 7

1. List every integer from −3 up to (but not including) 7.

AnswerA = {−3, −2, −1, 0, 1, 2, 3, 4, 5, 6}

(ii) x is a natural number less than 6

1. Natural numbers less than 6 are 1, 2, 3, 4 and 5.

AnswerB = {1, 2, 3, 4, 5}

(iii) the sum of its digits is 8

1. A two-digit number has a tens digit from 1 to 9 and a units digit from 0 to 9.
2. Let the tens digit run 1 to 8, with the units digit chosen so the two digits add to 8.

AnswerC = {17, 26, 35, 44, 53, 62, 71, 80}

(iv) prime number which is divisor of 60

1. Break 60 into primes: 60 = 2 × 2 × 3 × 5.
2. The only primes dividing 60 are 2, 3 and 5.

AnswerD = {2, 3, 5}

(v) all letters in the word TRIGONOMETRY

1. Write out T-R-I-G-O-N-O-M-E-T-R-Y and drop the repeats (T, R and O each appear twice).

AnswerE = {T, R, I, G, O, N, M, E, Y}

(vi) all letters in the word BETTER

1. Write out B-E-T-T-E-R and drop the repeats (T and E each appear twice).

AnswerF = {B, E, T, R}

Watch this explained “A few that catch people out”, 11:39 into Listing the members versus stating the property they share

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(i) {3, 6, 9, 12}

1. Each term is 3 times its position: 3×1, 3×2, 3×3, 3×4.
2. The list stops at 12, so n runs from 1 to 4.

Answer{x : x = 3n, n ∈ N and 1 ≤ n ≤ 4}

(ii) {2, 4, 8, 16, 32}

1. Each term is a power of 2: 2¹, 2², 2³, 2⁴, 2⁵.

Answer{x : x = 2ⁿ, n ∈ N and 1 ≤ n ≤ 5}

(iii) {5, 25, 125, 625}

1. Each term is a power of 5: 5¹, 5², 5³, 5⁴.

Answer{x : x = 5ⁿ, n ∈ N and 1 ≤ n ≤ 4}

(iv) {2, 4, 6, . . .}

1. The list never stops (the dots keep going), and each term is 2 times its position.

Answer{x : x = 2n, n ∈ N}

(v) {1, 4, 9, . . ., 100}

1. Each term is a perfect square: 1², 2², 3², and so on, and the list stops at 100 = 10².

Answer{x : x = n², n ∈ N and 1 ≤ n ≤ 10}

Watch this explained “Roster to property: two sentences, one set”, 8:28 into Listing the members versus stating the property they share

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(i) x is an odd natural number

1. Odd natural numbers go 1, 3, 5, 7, 9 and then keep adding 2 forever.
2. There is no last one, so the list is written with dots.

AnswerA = {1, 3, 5, 7, 9, …}

(ii) x is an integer, −1/2 < x < 9/2

1. −1/2 is −0.5 and 9/2 is 4.5.
2. The integers strictly between −0.5 and 4.5 are 0, 1, 2, 3, 4.

AnswerB = {0, 1, 2, 3, 4}

(iii) x is an integer, x² ≤ 4

1. x² ≤ 4 means x lies between −2 and 2, inclusive.
2. The integers in that range are −2, −1, 0, 1, 2.

AnswerC = {−2, −1, 0, 1, 2}

(iv) x is a letter in the word “LOYAL”

1. LOYAL has the letters L, O, Y, A, L − L repeats.

AnswerD = {L, O, Y, A}

(v) a month of a year not having 31 days

1. Of the twelve months, January, March, May, July, August, October and December have 31 days.
2. The other five do not.

AnswerE = {February, April, June, September, November}

(vi) a consonant in the English alphabet which precedes k

1. The letters before k are a, b, c, d, e, f, g, h, i, j.
2. Remove the vowels a, e and i from that list.

AnswerF = {b, c, d, f, g, h, j}

Watch this explained “Property to roster: derive, do not guess”, 7:19 into Listing the members versus stating the property they share

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(i) {1, 2, 3, 6}

1. 1, 2, 3 and 6 are exactly the natural-number divisors of 6.

AnswerMatches (c)

(ii) {2, 3}

1. 2 and 3 are the divisors of 6 that are also prime.

AnswerMatches (a)

(iii) {M,A,T,H,E,I,C,S}

1. M, A, T, H, E, I, C, S are the distinct letters of MATHEMATICS.

AnswerMatches (d)

(iv) {1, 3, 5, 7, 9}

1. 1, 3, 5, 7, 9 are exactly the odd natural numbers below 10.

AnswerMatches (b)

Watch this explained “The check has two halves”, 10:25 into Listing the members versus stating the property they share